decreasing sequence of nonempty closed sets in M
Solution 1:
Compactness is necessary and your argument is not valid. For a counter-example consider the real line with the usual metric. Let $F_n=(-\infty, -n]$ and $G=(0,\infty)$. Then $\bigcap F_n=\emptyset \subseteq G$ but no $F_n$ is contained in $G$.
Solution 2:
For a counter-example with a non-empty intersection:
Let $F_n=(-\infty,-n]\cup\{7\}$ and $G=(0,\infty)$. Then $\bigcap F_n=\{7\} \subseteq G$, but no $F_n$ is contained in $G$.